European Journal of Combinatorics 30 (2009) 1281–1288

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European Journal of Combinatorics

journal homepage: www.elsevier.com/locate/ejc

On arithmetic partitions of ZnVictor J.W. Guo a, Jiang Zeng ba Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of Chinab Université de Lyon; Université Lyon 1; Institut Camille Jordan, UMR 5208 du CNRS; 43, boulevard du 11 novembre 1918, F-69622Villeurbanne Cedex, France

a r t i c l e i n f o

Article history:Received 10 June 2008Received in revised form29 October 2008Accepted 25 November 2008Available online 24 February 2009

a b s t r a c t

Generalizing a classical problem in enumerative combinatorics,Mansour and Sun counted the number of subsets of Zn withoutcertain separations. Chen, Wang, and Zhang then studied theproblemof partitioningZn into arithmetical progressions of a giventype under some technical conditions. In this paper, we improve ontheir main theorems by applying a convolution formula for cyclicmultinomial coefficients due to Raney–Mohanty.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In his solution of problème desménagesKaplansky [1] showed that the number ofways of selecting kelements, no two consecutive, from n objects arrayed on a cycle is n

n−k

(n−kk

). LetZn := {0, 1, . . . , n−

1} be the set of congruence classesmodulo nwith usual arithmetic. Then Yamamoto [2] (see also [3, p.222]) proved that ifn > pk+1 the number ofways of selecting k elements fromZn, no two consecutive,is

nn− pk

(n− pkk

), (1.1)

when i± 1, . . . , i± p are regarded as consecutive to i.In the last three decades a lot of generalizations and variations of Kaplansky’s problem have been

studied by several authors (see, for example, [4–13]). In particular, Konvalina [4] considered thenumber of k-subsets {x1, x2, . . . , xk} of Zn such that xi − xj 6= 2 for all 1 6 i, j 6 k, and found that the

answer is nn−k

(n−kk

)if n > 2k+1. Hwang [7] then generalizedKonvalina’s result to the case xi−xj 6= m

E-mail addresses: jwguo@math.ecnu.edu.cn (V.J.W. Guo), zeng@math.univ-lyon1.fr (J. Zeng).URLs: http://math.ecnu.edu.cn/∼jwguo (V.J.W. Guo), http://math.univ-lyon1.fr/∼zeng (J. Zeng).

0195-6698/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ejc.2008.11.009

1282 V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288

and deduced that the desired number is given by the same formula if n > mk+ 1. Recently, Mansourand Sun [14] gave the following unification of Yamamoto’s and Hwang’s formulas.

Theorem 1.1 (Mansour–Sun). Let m, n, p, k be positive integers such that n > mpk+1. Then the numberof k-subsets {x1, x2, . . . , xk} of Zn such that

xi − xj 6∈ {m, 2m, . . . , pm} (1 6 i, j 6 k), (1.2)

is also given by (1.1).

A short proof of Theorem 1.1was given by Guo [15] by using Rothe’s identity. In order to generalizeMansour–Sun’s result, Chen,Wang, and Zhang [16] defined anm-AP-block of length k to be a sequence(x1, x2, . . . , xk) of distinct elements in Zn such that xi+1 − xi = m for 1 6 i 6 k − 1 and studied theproblem of partitioning Zn into m-AP-blocks. The type of such a partition is defined to be the type ofthe multiset of the lengths of the blocks. For example, the following is a 3-AP-partition of Z20 of type14233241:

(2), (4, 7), (5, 8), (6), (9, 12, 15), (10), (11), (13, 16, 19), (14, 17, 0, 3), (18, 1).We need to emphasize that (x, x+m, . . . , x+ (n− 1)m) and (x+m, x+ 2m, . . . , x+ (n− 1)m, x)and so on are deemed as different m-AP-blocks in Zmn. For example, all the 2-AP-partitions of Z6 oftype 32 are

{(i, i+ 2, i+ 4), (j+ 1, j+ 3, j+ 5)}i,j=0,2,4.Chen, Wang, and Zhang [16] constructed a bijection between m-AP-partitions and m′-AP-partitionsof Zn under some technical conditions, and established the following theorem.

Theorem 1.2 (Chen–Wang–Zhang). Let m, n, k1, k2, . . . , kr and i2, . . . , ir be positive integers such that1 < i2 < · · · < ir and

k1 > (k2 + · · · + kr) ((m− 1)(ir − 1)− 1) . (1.3)

Then the number of partitions of Zn into m-AP-blocks of type 1k1 ik22 · · · i

krr does not depend on m, and is

given by the cyclic multinomial coefficient

nk1 + · · · + kr

(k1 + · · · + krk1, . . . , kr

). (1.4)

If we specialize the type to 1n−(p+1)k(p + 1)k, then the condition (1.3) becomes n > mpk + 1.Furthermore, if (x1, x1 + m, . . . , x1 + pm), . . . , (xk, xk + m, . . . , xk + pm) are the k blocks of lengthp+ 1 in anm-AP-partition of Zn of type 1n−(p+1)k(p+ 1)k, then the set {x1, . . . , xk} satisfies (1.2), andvice versa. Therefore Theorem 1.2 implies Theorem 1.1.In this paper we shall improve and complete Theorem 1.2 by establishing the following two

theorems.

Theorem 1.3. Let m, n be positive integers, and let k1, k2, . . . , kr be nonnegative integers such thatn = k1 + 2k2 + · · · + rkr . Let d = gcd(m, n). If

∆ := n− d(n− k1 − · · · − kr) > 0, (1.5)

then the number of partitions of Zn into m-AP-blocks of type 1k12k2 · · · rkr is given by (1.4).

It is not hard to see that the condition (1.5) is weaker than (1.3), i.e., the condition (1.3) impliesthat (1.5). In other words, for fixed n and a given type, there are in general many more m’s satisfying(1.5) than satisfying (1.3). For example, by Theorem 1.3, the numbers of m-AP-partitions of Z120 oftype 18923325172 are all equal for

m = 1, 2, 3, 4, 5, 7, 9, 11, 13, 14, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31,33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59,

i.e., for d = 1, 2, 3, 4, 5. However, Theorem 1.2 only asserts that these numbers for m = 1, 2, 3 areequal.

V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288 1283

Theorem 1.4. Let k1, k2, . . . , kr ,m, n, d and∆ be given as in Theorem 1.3. Then the number of partitionsof Zn into m-AP-blocks of type 1k12k2 · · · rkr is given by

nk1 + · · · + kr

(k1 + · · · + krk1, . . . , kr

)+n(−1)k2+···+kr

k2 + · · · + kr

(k2 + · · · + krk2, . . . , kr

), if ∆ = 0,

nk1 + · · · + kr

(k1 + · · · + krk1, . . . , kr

)+♣(−1)k2+···+kr

(k2 + · · · + krk2, . . . , kr

), if ∆ = −d,

where

♣ =

n, if k2 = 0,

n(1−

n(1− d−1)k2(k2 + · · · + kr)(k2 + · · · + kr − 1)

), if k2 > 0.

When the type in Theorem 1.4 is 1n−(p+1)k(p+ 1)k again, then∆ = n−mpk. To assure that thereis anm-AP-block of length p+ 1 in Zn, we need to assume that n > pm, which is equivalent to k > 2if ∆ = 0 and pk > p + 1 if ∆ = −m. As mentioned after Theorem 1.2, each family of km-AP-blocksin Zn is in one-to-one correspondence with a k-subset of Zn satisfying (1.2), we derive the followingtwo results, which can be viewed as complements to Theorem 1.1.

Corollary 1.5. Let m, p > 1, k > 2 and n = mpk. Then the number of k-subsets {x1, x2, . . . , xk} of Znsuch that xi − xj 6∈ {m, 2m, . . . , pm} for all 1 6 i, j 6 k, is given by

nn− pk

(n− pkk

)+ (−1)k

nk. (1.6)

Actually the above formula is deduced for m > 2, i.e., n > (p + 1)k, but it also holds for m = 1 ifwe take the convention

limx→0

nx

( xk

)= (−1)k−1

nk,

and so (1.6) is equal to 0 in this case. Here is an example for Corollary 1.5. For m = p = k = 2, thenumber of 2-subsets {x1, x2} of Z8 such that x1 − x2, x2 − x1 6∈ {2, 4} is equal to

84

(42

)+ 4 = 16,

and the corresponding subsets are {i, i+ 1} and {i, i+ 3}, where i ∈ Z8.

Corollary 1.6. Let m, p, k > 1 with pk > p + 1 and let n = mpk − m. Then the number of k-subsets{x1, x2, . . . , xk} of Zn such that xi − xj 6∈ {m, 2m, . . . , pm} for all 1 6 i, j 6 k, is given by

nn− k

(n− kk

)+ (−1)k−1n(m− 2), if p = 1,

nn− pk

(n− pkk

)+ (−1)kn, if p > 2.

Similarly, although the above formula is deduced for mpk − m > (p + 1)k, it also holds withoutthis condition. The details are left to the interested reader.

Remark. For 0 < m < n, let gm(n, k) denote the number of k-subsets {x1, x2, . . . , xk} of Zn such thatxi − xj 6= m for all 1 6 i, j 6 k. Hwang [7, Corollary 2] obtained

gm(n, k) =bd/2c∑j=0

(−1)nj/d(dj

)n− 2nj/dn− k− nj/d

(n− k− nj/dk− nj/d

), (1.7)

1284 V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288

Fig. 1. A 20-cycle dissection of type 14233241 .

where d = gcd(m, n). Letting n = mk or n = mk − m in (1.7), we are led to the p = 1 case ofCorollary 1.5 or 1.6. However, since there are two cases in Corollary 1.6, it seems impossible to give aformula like (1.7) to unify Corollaries 1.5 and 1.6 for general p.We recall and establish some necessary lemmas in Section 2 and prove Theorems 1.3 and 1.4 in

Sections 3 and 4, respectively. Our main idea is the following: Lemma 2.4 permits us to reduce thegeneralm-AP-partition problem of Zn to the case wherem divides n. For the latter we may write thepartition number as a multiple sum, which can be computed by applying Raney–Mohanty’s identity.

2. Some lemmas

A dissection of an n-cycle is a 1-AP-partition ofZn, which can be depicted by inserting a bar betweenany two consecutive blocks on an n-cycle. For example, Fig. 1 illustrates a 20-cycle dissection of type14233241.It is easy to see that the number of dissections of Zn is given by (1.4). Indeed, deleting the segment

containing 0 in any dissection of n-cycle of type 1k12k2 · · · rkr yields a dissection of a (n − i)-line oftype 1k1 · · · iki−1 · · · rkr if the segment containing 0 is of length i (1 6 i 6 n). So the number of suchdissections of n-cycle is equal to

i(k1 + · · · + (ki − 1)+ · · · + krk1, . . . , ki − 1, . . . , kr

). (2.1)

Summing (2.1) over all i yields the following known result (see [17, Lemma 3.1]).

Lemma 2.1 (Chen–Lih–Yeh). For an n-cycle, the number of dissections of type 1k12k2 · · · rkr is given bythe cyclic multinomial coefficient (1.4).

For any variable x and nonnegative integers k1, . . . , kr define the multinomial coefficient(x

k1, k2, . . . , kr

):=x(x− 1) . . . (x− k1 − · · · − kr + 1)

k1!k2! . . . kr !.

Note that when x = k1 + k2 + · · · + kr the above definition coincides with the classical definition ofmultinomial coefficient and(

k1 + · · · + krk1, . . . , kr

)=

(k1 + · · · + krk2, . . . , kr

).

The following convolution formula for multinomial coefficients is due to Raney–Mohanty [18,19].For other proofs of (2.2), we refer the reader to [20–22].

V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288 1285

Lemma 2.2 (Raney–Mohanty’s identity). For any variables x, y, z1, . . . , zm and nonnegative integersN1, . . . ,Nm, there holds∑

06ti6Nii=1,...,m

xx− t1z1 − · · · − tmzm

(x− t1z1 − · · · − tmzm

t1, . . . , tm

)

×y

y− (N1 − t1)z1 − · · · − (Nm − tm)zm

(y− (N1 − t1)z1 − · · · − (Nm − tm)zm

N1 − t1, . . . ,Nm − tm

)=

x+ yx+ y− N1z1 − · · · − Nmzm

(x+ y− N1z1 − · · · − Nmzm

N1, . . . ,Nm

). (2.2)

We also need the following elementary arithmetical result (see [23, Theorem 5.32 and Exercise 16on page 127] or [15]).

Lemma 2.3. Let m, n be positive integers. If gcd(m, n) = d, then there exists an integer a such thatgcd(a, n) = 1 and am ≡ d(mod n).

The following is our key lemma.

Lemma 2.4. If m, n > 1 and gcd(m, n) = d, then there is a bijection from the set of m-AP-partitions ofZn to the set of d-AP-partitions of Zn. Moreover this bijection keeps the type of partitions.

Proof. By Lemma 2.3, there exists an inversible element a ∈ Zn such that am = d. Let a−1 be theinverse of a. For any subset B of Zn and x ∈ Zn, let xB = {xb: b ∈ B}. If {B1, B2, . . . , Bs} is an m-AP-partition of Zn, then {aB1, aB2, . . . , aBs} is a d-AP-partition of Zn. Conversely, if {C1, C2, . . . , Cs}is a d-AP-partition of Zn, then {a−1C1, a−1C2, . . . , a−1Cs} is an m-AP-partition of Zn. Obviously, thiscorrespondence keeps the type of partitions. This proves the lemma. �

It follows from Lemma 2.4 that if there exists anm-AP-partition of Zn of a given type 1k12k2 · · · rkr(kr > 0) then gcd(m, n)r 6 n.

3. Proof of Theorem 1.3

By Lemma 2.4, it suffices to consider the case where m divides n, i.e., d = m. Let n = mn1 anddivide Zn intom subsets of the same cardinality n1:

Zn,j = {mi+ j: i = 0, . . . , n1 − 1}, 0 6 j 6 m− 1.

Hence Zn =⊎m−1j=0 Zn,j. Let B = {B1, B2 . . . , Bs} be an m-AP-partition of Zn of type 1k12k2 · · · rkr

(r 6 n1). Then Bi,j = Zn,j ∩ Bi is equal to ∅ or Bi for 1 6 i 6 s and 0 6 j 6 m − 1. Furthermore, sincethe transformation x 7→ (x− j)/mmaps eachm-AP-block Bi,j of Zn,j (0 6 j 6 m− 1) to a 1-AP-blockB′i,j of Zn1 , each m-AP-partition Bj = {B1,j, . . . , Bs,j} corresponds bijectively to a 1-AP-partition B ′j ofZn1 with the same type. Thus, we have established a bijection between the set of m-AP-partitions ofZn and the set ofm-tuples of 1-AP-partitions of Zn1 :B ↔ (B ′0, . . . ,B

′

m−1).Now assume that the m-AP-partition B is of type 1k12k2 · · · rkr (r 6 n1), and the corresponding

1-AP-partitionB ′j is of type 1k1,j2k2,j · · · rkr,j (0 6 j 6 m− 1). Clearly,

k2,0 + k2,1 + · · · + k2,m−1 = k2,k3,0 + k3,1 + · · · + k3,m−1 = k3,· · ·

kr,0 + kr,1 + · · · + kr,m−1 = kr .

(3.1)

1286 V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288

By Lemma 2.1 and noticing that n1 = k1,j+ 2k2,j+ · · · + rkr,j, the number of 1-AP-partitions of Zn1 oftype 1k1,j2k2,j · · · rkr,j is equal to

n1k1,j + k2,j + · · · + kr,j

(k1,j + k2,j + · · · + kr,jk1,j, k2,j, . . . , kr,j

)=

n1n1 − k2,j − · · · − (r − 1)kr,j

(n1 − k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

).

For m, n > 1 let fm,n(k1, . . . , kr) be the number of partitions of Zn into m-AP-blocks of type1k12k2 · · · rkr . Then

fm,n(k1, . . . , kr) =∑(ki,j)

m−1∏j=0

n1n1 − k2,j − · · · − (r − 1)kr,j

(n1 − k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

), (3.2)

where the summation is over allmatrices (ki,j) 26i6r06j6m−1

of nonnegative integral coefficients ki,j satisfying(3.1) and

n1 − k2,0 − · · · − (r − 1)kr,0 > 0,n1 − k2,1 − · · · − (r − 1)kr,1 > 0,· · ·

n1 − k2,m−1 − · · · − (r − 1)kr,m−1 > 0.

(3.3)

Recall that

∆ = n−m(n− k1 − · · · − kr) = mn1 −m(k2 + · · · + (r − 1)kr).

If∆ > 0, then we have n1 > k2 + · · · + (r − 1)kr , and thus all nonnegative integral solutions to (3.1)also satisfy (3.3) as ki > ki,j (2 6 i 6 r , 0 6 j 6 m− 1).It remains to prove that the right-hand side of (3.2) is equal to (1.4), namely

mn1mn1 − k2 − · · · − (r − 1)kr

(mn1 − k2 − · · · − (r − 1)kr

k2, . . . , kr

). (3.4)

We proceed by induction on m > 1. This is equivalent to repeatedly applying Raney–Mohanty’sidentity (2.2). The case m = 1 is obviously true. Suppose that the formula is true for m − 1 withm > 2 and let ki,0 + ki,1 + · · · + ki,m−2 = k′i be fixed for i = 2, . . . , r . Then∑

ki,0,...,ki,m−2i=2,...,r

m−2∏j=0

n1n1 − k2,j − · · · − (r − 1)kr,j

(n1 − k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

)

=(m− 1)n1

(m− 1)n1 − k′2 − · · · − (r − 1)k′r

((m− 1)n1 − k′2 − · · · − (r − 1)k

′r

k′2, . . . , k′r

).

Plugging this into (3.2) yields

fm,n(k1, . . . , kr)

=

∑k′i+ki,m−1=kii=2,...,r

(m− 1)n1(m− 1)n1 − k′2 − · · · − (r − 1)k′r

((m− 1)n1 − k′2 − · · · − (r − 1)k

′r

k′2, . . . , k′r

)

×n1

n1 − k2,m−1 − · · · − (r − 1)kr,m−1

(n1 − k2,m−1 − · · · − (r − 1)kr,m−1

k2,m−1, . . . , kr,m−1

)which is (3.4) by applying Raney–Mohanty’s identity (2.2).

V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288 1287

4. Proof of Theorem 1.4

For the case∆ = 0 or∆ = −m, the number fm,n(k1, . . . , kr) is again given by (3.2). However, wewill meet with n1 − k2,j − · · · − (r − 1)kr,j 6 0 for some 0 6 j 6 m− 1 in some nonnegative integralsolutions (ki,j) to (3.1). It is convenient here to consider a more general form of (3.2) as follows. Forany variable x, let fm,n(x; k1, . . . , kr) be the following expression∑

(ki,j)

m−1∏j=0

xx− k2,j − · · · − (r − 1)kr,j

(x− k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

),

where (ki,j) ranges over the same integral matrices as (3.2).Let M be the set of all nonnegative integral matrices (ki,j) 26i6r

06j6m−1satisfying (3.1), and let S be the

set of all (ki,j) inM such that (3.3) does not hold. Then

fm,n(x; k1, . . . , kr)

=

∑(ki,j)∈M

m−1∏j=0

xx− k2,j − · · · − (r − 1)kr,j

(x− k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

)

−

∑(ki,j)∈S

m−1∏j=0

xx− k2,j − · · · − (r − 1)kr,j

(x− k2,j − · · · − (r − 1)kr,j

k2,j, . . . , kr,j

). (4.1)

When∆ = 0, we have n1 = k2 + 2k3 + · · · + (r − 1)kr , and S reduces to

S1 :={(ki,j): for some j0 and all i, we have ki,j0 = ki and ki,j = 0 if j 6= j0

}.

So the second summation on the right-hand side of (4.1) becomes

mxx− n1

(x− n1k2, . . . , kr

),

while the first summation can be summed by using Raney–Mohanty’s identity. It follows that

fm,n(x; k1, . . . , kr)

=mx

mx− k2 − · · · − (r − 1)kr

(mx− k2 − · · · − (r − 1)kr

k2, . . . , kr

)−

mxx− n1

(x− n1k2, . . . , kr

). (4.2)

Letting x = n1 in (4.2) and noticing the following fact

limz→0

1z

(z

a1, . . . , as

)=(−1)a1+···+as−1

a1 + · · · + as

(a1 + · · · + asa1, . . . , as

), (4.3)

one obtains the first formula in Theorem 1.4.When ∆ = −m, we have n1 = k2 + 2k3 + · · · + (r − 1)kr − 1. If k2 = 0, then S = S1, while if

k2 > 0, then

S = S1 ∪ {(ki,j): for some j0 6= j1, we have k2,j0 = k2 − 1, k2,j1 = 1,ki,j0 = ki(2 < i 6 r) and ki,j = 0 otherwise }.

It follows that

fm,n(x; k1, . . . , kr)

=mx

mx− k2 − · · · − (r − 1)kr

(mx− k2 − · · · − (r − 1)kr

k2, . . . , kr

)−

mxx− n1 − 1

(x− n1 − 1k2, . . . , kr

)− χ(k2 > 0)

m(m− 1)x2

x− n1

(x− n1

k2 − 1, k3, . . . , kr

). (4.4)

1288 V.J.W. Guo, J. Zeng / European Journal of Combinatorics 30 (2009) 1281–1288

Letting x = n1 in (4.4) and using (4.3) and(−1

a1, . . . , as

)= (−1)a1+···+as

(a1 + · · · + asa1, . . . , as

),

we obtain the second formula in Theorem 1.4.

Remark. It is also possible to compute fm,n(k1, . . . , kr) for the case ∆ = −2 gcd(m, n) or ∆ =−3 gcd(m, n). But the result is more complicated and is omitted here.

Acknowledgments

We thank the referees for helpful comments on a previous version of this paper. This work wasdone during the first author’s visit to Institut Camille Jordan of Univerité Lyon I, andwas supported byProjectMIRA 2007 de la Région Rhône-Alpes. The first authorwas also supported by Shanghai LeadingAcademic Discipline Project, Project Number: B407.

References

[1] I. Kaplansky, Solution of the Problème des ménages, Bull. Amer. Math. Soc.. 49 (1943) 784–785.[2] K. Yamamoto, Structure polynomial of Latin rectangles and its application to a combinatorial problem, Mem. Fac. Sci.Kyusyu Univ. Ser. A 10 (1956) 1–13.

[3] J. Riordan, An Introduction to Combinatorial Analysis, Wiley, New York, 1958.[4] J. Konvalina, On the number of combinations without unit separation, J. Combin. Theory Ser. A 31 (1981) 101–107.[5] W. Chu, On the number of combinations without k-separations, J. Math. Res. Exposition 7 (1987) 511–520 (in Chinese).[6] F.K. Hwang, Cycle polynomials, Proc. Amer. Math. Soc.. 83 (1981) 215–219.[7] F.K. Hwang, Selecting k objects from a cycle with p pairs of separation s, J. Combin. Theory Ser. A 37 (1984) 197–199.[8] F.K. Hwang, J. Korner, V.K.-W. Wei, Selecting non-consecutive balls arranged in many lines, J. Combin. Theory Ser. A 37(1984) 327–336.

[9] P. Kirschenhofer, H. Prodinger, Two selection problems revisited, J. Combin. Theory Ser. A 42 (1986) 310–316.[10] W.O.J. Moser, The number of subsets without a fixed circular distance, J. Combin. Theory Ser. A 43 (1986) 130–132.[11] H. Prodinger, On the number of combinations without a fixed distance, J. Combin. Theory Ser. A 35 (1983) 362–365.[12] M.A. Shattuck, C.G. Wagner, A new statistic on linear and circular r-mino arrangements, Electron. J. Combin. 13 (2006)

#R42.[13] E. Munarini, N.Z. Salvi, Scattered subsets, Discrete Math. 267 (2003) 213–228.[14] T.Mansour, Y. Sun, On thenumber of combinationswithout certain separations, European J. Combin. 29 (2008) 1200–1206.[15] V.J.W. Guo, A new proof of a theorem of Mansour and Sun, European J. Combin. 29 (2008) 1582–1584.[16] W.Y.C. Chen, D.G.L. Wang, I.F. Zhang, Partitions of Zn into arithmetic progressions, European J. Combin. (2008),

doi:10.1016/j.ejc.2008.09.027.[17] W.Y.C. Chen, K.W. Lih, Y.N. Yeh, Cyclic tableaux and symmetric functions, Stud. Appl. Math. 94 (1995) 327–339.[18] G.N. Raney, Functional composition patterns and power series reversion, Trans. Amer. Math. Soc.. 94 (1960) 441–451.[19] S.G. Mohanty, Some convolutions with multinomial coefficients and related probability distributions, SIAM Rev. 8 (1966)

501–509.[20] V.J.W. Guo, Bijective proofs of Gould–Mohanty’s and Raney–Mohanty’s identities, Ars Combin. (in press).[21] V. Strehl, Identities of Rothe–Abel–Schläfli–Hurwitz-type, Discrete Math. 99 (1992) 321–340.[22] J. Zeng, Multinomial convolution polynomials, Discrete Math. 160 (1996) 219–228.[23] T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, Heidelberg, 1976.